Solving the boundary value problems of the heat equation in noncylindrical domains degenerating at the initial moment leads to the necessity of research of the singular Volterra integral equations of the second kind, when the norm of the integral operator is equal to 1. The paper deals with the singular Volterra integral equation of the second kind, to which by virtue of ‘the incompressibility’ of the kernel the classical method of successive approximations is not applicable. It is shown that the corresponding homogeneous equation when \(|\lambda|>1\) has a continuous spectrum, and the multiplicity of the characteristic numbers increases depending on the growth of the modulus of the spectral parameter \(|\lambda|\). By the Carleman-Vekua regularization method (Vekua in Generalized Analytic Functions, 1988) the initial equation is reduced to the Abel equation. The eigenfunctions of the equation are found explicitly. Similar integral equations also arise in the study of spectral-loaded heat equations (Amangaliyeva et al. in Differ. Equ. 47(2):231-243, 2011).

Investigation of boundary value problems for the heat equation in noncylindrical domains has wide practical application [1–3]. For example, in the study of thermal regimes of the various electrical contacts there is the necessity to study the processes of heat and mass transfer taking place between the electrodes. After achieving the melting temperature at the contact surface of electrodes there is a liquid metal bridge between these electrodes. When the contacts open this bridge is divided into two parts, i.e. the contact material is transferred from one electrode to another, and this leads to the bridging erosion. Ultimately, the smooth surface of contacts is destroyed, which means that their proper operation is violated. The mathematical description of the thermal processes which go with the bridging erosion, leads to solving the boundary value problems for the heat equation in domains with moving boundary, namely in the domains which degenerate into a point at the initial moment. Using the apparatus of heat potentials, solving the problems under consideration is reduced to the study of singular Volterra integral equations of the second kind, when the norm of the integral operator is equal to 1. A feature of these equations is the incompressibility of the kernel and this is expressed in the fact that the corresponding nonhomogeneous equation cannot be solved by classical methods.

For the problem of the solvability of the Volterra integral equation of the second kind with a special kernel, stated in Section 2, after some transformation in Section 3 we obtain the corresponding characteristic integral equation. An important moment of our research is fact that using a Carleman-Vekua regularization method [4], we reduce the initial problem to solving the Abel integral equation of the second kind. The solution of the last equation provides finding all solutions of the initial integral equation from Section 2. These results are stated in Sections 4-6. The main result about solvability of the integral equation in a class of essentially bounded functions is formulated in the form of the theorem in Section 6.

Problem

To find the solution \(\varphi(t)\) of integral equation (1) satisfying the condition \(\sqrt{t}\cdot \varphi(t)\in L_{\infty}(0,\infty)\) for any given function \(\sqrt{t}\cdot f(t)\in L_{\infty}(0,\infty)\) and each given complex spectral parameter \(\lambda\in\mathcal{C}\).

We note that the integral equations of the form (1) arise in the study of boundary value problems of heat conduction in an infinite angular domain, which degenerates at the initial moment. Such equations are called by us Volterra integral equations with ‘incompressible’ kernel. The feature of the equation in question consists in property (3) of the kernel \(K(t, \tau)\) and is expressed in the fact that the corresponding nonhomogeneous equation cannot be solved by the method of successive approximations for \(|\lambda|>1\). Obviously, if \(|\lambda|<1\), then (1) has a unique solution, which can be found by the method of successive approximations. The case \(\lambda=1\) was considered in [5], where it is shown that (1) has only one nontrivial solution at \(f(t)\equiv0\) (within a constant factor). Further in this paper, we assume that \(|\lambda|>1\).

The equations of the form (1) were first considered by SN Kharin: the asymptotics of integrals of the double layer potentials were studied, and approximate solutions of some applied problems were constructed [6, 7]. Subsequently such integral equations were the subject of investigation by many authors.

It should be noted that the boundary value problems for spectrally loaded parabolic equations also are reduced to the singular integral equations under consideration when the load line moves by the law \(x=\alpha(t)\) [8, 9].

where \(\delta(x)\) is the delta-function, \(C_{k}=\mathrm{const}\), and \(p_{k} \) (\(k=0,\pm1, \pm2,\ldots\)) are roots of (14). Applying to the last equality the inverse Laplace transformation, we obtain

In the last integral we have carried out the integration along the contour, avoiding the points \(p_{k}\), determined by (16), on the left. The integral is understood in the sense of the Cauchy principal value. Since we consider \(y\leq0\), we close on the right cutting the half-plane (slit is along the positive real semiaxis). The zeros of the denominator of the function

We find the solution of the Abel equation (23) for \(\tilde{f}_{2}(t)=0\), that is, we will find a solution of the corresponding homogeneous equation (4) for each k, \(-N_{1}\leq k\leq N_{2}\) (eigenfunctions). Under this condition, (23) for each k, \(-N_{1}\leq k\leq N_{2}\), has the form

is a solution of the Abel equation (23) for \(\tilde{f}_{2}(t)=0\), that is, a solution of the ‘simplified’ homogeneous equation (4), and the functions \(\tilde{\varphi}_{k}(t)\) and values \(p_{k}\) are determined by (27) and (16), respectively.

We note that after multiplying equality (28) by \(\exp (-t/(4a^{2}) )\), we obtain the solution of the homogeneous equation corresponding to the original equation (1):

Here it is sufficient to take into account that the numbers \(p_{k}\), \(k\in[-N_{1}, N_{2}]\), are the roots of (14) for each fixed complex spectral parameter \(\lambda\in\mathcal{C}\), and to use the asymptotic form of the function \(\operatorname{erfc}(z)\) for large values of z ([13], p.890, formula 8.2548; [14], p.758). Obviously, there is a limit relation

We studied the problems of resolvability of singular Volterra integral equations of the second kind in the space of essentially bounded functions. It is proved that at \(|\lambda|>1\) the homogeneous equation which corresponds to (1) has a continuous spectrum, and the multiplicity of the characteristic numbers increases depending on the growth of the modulus of the spectral parameter \(|\lambda|\). The initial equation (1) is reduced to the Abel integral equation (23) by the regularization method of Carleman-Vekua [4], which was developed for solving singular integral equations. The eigenfunctions of (1) are found explicitly and their multiplicity depending on the modulus of the characteristic number \(|\lambda|\) is found.

Acknowledgements

This study was financially supported by Committee of Science of the Ministry of Education and Sciences (Grant 0112 RK 00619/GF on priority ‘Intellectual potential of the country’).

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.